Scale-dependent Measure Research Articles

RMS asymmetry: a robust metric of galaxy shapes in images with varied depth and resolution

Structural disturbances, such as galaxy mergers or instabilities, are key candidates for driving galaxy evolution, so it is important to detect and quantify galaxies hosting these disturbances spanning a range of masses, environments, and cosmic times. Traditionally, this is done by quantifying the asymmetry of a galaxy as part of the concentration-asymmetry-smoothness system, , and selecting galaxies above a certain threshold as merger candidates. However, in this work, we show that , is extremely dependent on imaging properties – both resolution and depth – and thus defining a single threshold is impossible. We analyze an alternative root-mean-squared asymmetry, , and show that it is independent of noise down to the average SNR per pixel of 1. However, both metrics depend on the resolution. We argue that asymmetry is, by design, always a scale-dependent measurement, and it is essential to define an asymmetry at a given physical resolution, where the limit should be defined by the size of the smallest features one wishes to detect. We measure asymmetry of a set of z≈0.1 galaxies observed with HST, HSC, and SDSS, and show that after matching the resolution of all images to 200 pc, we are able to obtain consistent measurements with all three instruments despite the vast differences in the original resolution or depth. We recommend that future studies use measurement when evaluating asymmetry, where x is defined by the physical size of the features of interest, and is kept consistent across the dataset, especially when the redshift or image properties of galaxies in the dataset vary.

Scale and Time Dependence of Alfvénicity in the Solar Wind as Observed by the Parker Solar Probe

Alfvénicity is a well-known property, common in the solar wind, characterized by a high correlation between magnetic and velocity fluctuations. Data from the Parker Solar Probe (PSP) enable the study of this property closer to the Sun than ever before, as well as in the sub-Alfvénic solar wind. We consider scale-dependent measures of Alfvénicity based on second-order functions of the magnetic and velocity increments as a function of time lag, including the normalized cross helicity σ c and residual energy σ r . Scale-dependent Alfvénicity is strongest for lags near the correlation scale and increases when moving closer to the Sun. We find that σ r typically remains close to the maximally negative value compatible with σ c . We did not observe significant changes in measures of Alfvénicity between sub-Alfvénic and super-Alfvénic wind. During most times, the solar wind was highly Alfvénic; however, lower Alfvénicity was observed when PSP approached the heliospheric current sheet or other magnetic structures with sudden changes in the radial magnetic field, non-unidirectional strahl electron pitch angle distributions, and strong electron density contrasts. These results are consistent with a picture in which Alfvénic fluctuations generated near the photosphere transport outward, forming highly Alfvénic states in the young solar wind, and subsequent interactions with large-scale structures and gradients lead to weaker Alfvénicity, as commonly observed at larger heliocentric distances.

Adaptive Wavelet Domain Principal Component Analysis for Nonstationary Time Series

High-dimensional multivariate nonstationary time series, that is, data whose second order properties vary over time, are common in many scientific and industrial applications. In this article we propose a novel wavelet domain dimension reduction technique for nonstationary time series. By constructing a time-scale adaptive principal component analysis of the data, our proposed method is able to capture the salient dynamic features of the multivariate time series. We also introduce a new time and scale dependent cross-coherence measure to quantify the extent of association between a multivariate nonstationary time series and its proposed wavelet domain principal component representation. Theoretical results establish that our associated estimation scheme enjoys good bias and consistency properties when determining wavelet domain principal components of input data. The proposed method is illustrated using extensive simulations and we demonstrate its applicability on a real-world dataset arising in a neuroscience study. Supplementary materials, with proofs of theoretical results, additional simulations and code, are available online.

Measuring small-scale plasma irregularities in the high-latitude E- and F-regions simultaneously

The ionosphere, Earth’s space environment, exhibits widespread turbulent structuring, or plasma irregularities, visualized by the auroral displays seen in Earth’s polar regions. Such plasma irregularities have been studied for decades, but plasma turbulence remains an elusive phenomenon. We combine scale-dependent measurements from a ground-based radar with satellite observations to characterize small-scale irregularities simultaneously in the bottomside and topside ionosphere and perform a statistical analysis on an aggregate from both instruments over time. We demonstrate the clear mapping of information vertically along the ionospheric altitude column, for field-perpendicular wavelengths down to 1.5 km. Our results paint a picture of the northern hemisphere high-latitude ionosphere as a turbulent system that is in a constant state of growth and decay; energy is being constantly injected and dissipated as the system is continuously attempting an accelerated return to equilibrium. We connect the widespread irregularity dissipation to Pedersen conductance in the E-region, and discuss the similarities between irregularities found in the polar cap and in the auroral region in that context. We find that the effects of a conducting E-region on certain turbulent properties (small-scale spectral index) is near ubiquitous in the dataset, and so we suggest that the electrodynamics of a conducting E-region must be considered when discussing plasma turbulence at high latitudes. This intimate relationship opens up the possibility that E-region conductivity is associated with the generation of F-region irregularities, though further studies are needed to assess that possibility.

Seamount coral reefs are egg case nurseries for deep-sea skates.

Egg case nurseries of the boreal skate (Amblyraja hyperborea) and Richardson's skate (Bathyraja richardsoni) were defined and mapped on a bathyal seascape (~500-1,900 m depths) south of Tasmania, Australia, using 99 towed-camera transects (157 linear km; N= 50,858 images). In total, 738 skate egg cases were observed (present in 240 images, absent in 50,618); among 113 egg cases examined to identify parent species, 70% were A. hyperborea, 10% B. richardsoni and 20% unidentified Bathyraja species. 'Recently laid' egg cases were differentiated from 'aged' by classifying their colour and condition. The great majority (98%) of egg cases were observed in ~1,100-1,400 m depths on seamounts (15 of 36 surveyed), not seamount bases or adjacent continental slope. Egg cases were associated with reefs formed by accumulated skeletal matrix of the stony coral Solenosmilia variabilis, with >90% egg cases (including most of those recently laid) observed on living S. variabilis that characterises a 'coral zone' in ~950-1,350 m depths. Water in the coral zone is warmer (+0.66 to 2.37°C) than at the deep distributional limits of adult A. hyperborea and B. richardsoni (2,000 and 3,000 m, respectively), potentially providing for accelerated embryonic development. Co-occurrence with living coral infers an energetically favourable local-scale hydrodynamic environment for egg cases, particularly on seamount peaks, where increased water flow over egg cases would avert smothering by suspended sediment, and compensate for lower oxygen concentration compared to deeper depths occupied by adult skates. Criteria identifying egg case nurseries are strongly met for A. hyperborea at Seamount Z110 (468 egg cases of varied ages, maximum density of 5.47 m-2 ) and to a lesser extent on five others (Seamounts K1, Z16, Hill U, Z5 and Hill V). An abundance (density) criterion for defining nurseries needs to be flexible because it is a spatially scale-dependent measure that differs between surveys according to the tools and design employed. Off Australia, coral reef egg case nursery habitat is restricted to a narrow depth range in temperate latitudes where it is scarce and impacted by historical bottom trawl fishing in many locations. There has been effective conservation of nursery habitat, however, because four of the six nursery sites identified here and extensive coral reef areas are protected within marine parks. This article is protected by copyright. All rights reserved.

A scale-dependent measure of system dimensionality

SummaryA fundamental problem in science is uncovering the effective number of degrees of freedom in a complex system: its dimensionality. A system’s dimensionality depends on its spatiotemporal scale. Here, we introduce a scale-dependent generalization of a classic enumeration of latent variables, the participation ratio. We demonstrate how the scale-dependent participation ratio identifies the appropriate dimension at local, intermediate, and global scales in several systems such as the Lorenz attractor, hidden Markov models, and switching linear dynamical systems. We show analytically how, at different limiting scales, the scale-dependent participation ratio relates to well-established measures of dimensionality. This measure applied in neural population recordings across multiple brain areas and brain states shows fundamental trends in the dimensionality of neural activity—for example, in behaviorally engaged versus spontaneous states. Our novel method unifies widely used measures of dimensionality and applies broadly to multivariate data across several fields of science.

Predictive study of tuberculosis incidence by time series method and Elman neural network in Kashgar, China

ObjectivesKashgar, located in Xinjiang, China has a high incidence of tuberculosis (TB) making prevention and control extremely difficult. In addition, there have been very few prediction studies on TB incidence...

Scale-dependent measure of network centrality from diffusion dynamics

Classic measures of graph centrality capture distinct aspects of node importance, from the local (e.g., degree) to the global (e.g., closeness). Here we exploit the connection between diffusion and geometry to introduce a multiscale centrality measure. A node is defined to be central if it breaks the metricity of the diffusion as a consequence of the effective boundaries and inhomogeneities in the graph. Our measure is naturally multiscale, as it is computed relative to graph neighbourhoods within the varying time horizon of the diffusion. We find that the centrality of nodes can differ widely at different scales. In particular, our measure correlates with degree (i.e., hubs) at small scales and with closeness (i.e., bridges) at large scales, and also reveals the existence of multi-centric structures in complex networks. By examining centrality across scales, our measure thus provides an evaluation of node importance relative to local and global processes on the network.

Scale effects in remotely sensed greenspace metrics and how to mitigate them for environmental health exposure assessment

Metrics representing exposure to the natural environment are widely used in environmental health-related studies. They are calculated using a variety of different data sources representing greenspace and a range of buffer sizes representing human interaction with the environment. Previous studies have identified issues relating to buffer distance and scaling effects on greenspace exposure assessments when using satellite image-derived metrics. We evaluate the spatial scale sensitivity of three common greenspace metrics (i.e., Normalised Difference Vegetation Index- NDVI, Leaf Area Index- LAI, and Land Use and Land Cover-LULC), using lacunarity analysis, as a scale-dependent measure of heterogeneity based on the principles of fractals. By producing a ‘lacunarity curve’ across multiple spatial scales, we defined the scale-variances for specific greenspace metrics, including the upper scale limit at which the metrics become invariant, approximately 640 m for Sentinel-2 and 480 m for Landsat-8. Each of the greenspace metrics we considered exhibited scale sensitivities, meaning that each is expected to have a different influence on the strength and significance of the statistical associations found between greenspace exposure and health depending on the spatial scale of analysis (e.g., buffer distance). Using lacunarity curves, we produced a novel composite, multi-scale greenspace ‘exposure index’ in which each input scale is weighted according to its relative scale sensitivity. We also created a multi-scale, multi-metric map combining the different vegetation measures while accounting for scale. We found that cumulative exposure gradients across a large urban conurbation are even more marked when using our multi-scale ‘exposure index’ maps compared to traditional approaches. Our multi-scale, composite greenspace ‘exposure index’ mapping techniques are not as vulnerable to scale effects as traditional approaches and can be readily transferred to the analysis of other environmental exposure variables such as air pollution.

Zonal lacunarity analysis: a new spatial analysis tool for geographic information systems

Lacunarity as a scale-dependent measure of spatial heterogeneity has received great attention in landscape ecology. Most lacunarity measures have been obtained from greyscale or binary (0 and 1) data for an entire study area or fixed rectangular windows, and a zonal lacunarity tool for discrete raster data is still lacking in current geographic information systems. This short communication presents the development of a free zonal lacunarity analysis tool for ArcGIS to support applications involving scale-dependent analysis of spatial heterogeneity, including landscape ecology. The application of the tool is also demonstrated using 2001 and 2011 land cover data from the National Land Cover Database (NLCD). Based on the gliding-box algorithm for lacunarity estimation, a tool for zonal lacunarity analysis of discrete raster data is developed using ArcPy and the Python programming language. The tool uses discrete raster data as input, an optional zone feature class as zone data to partition the input raster data into different zones, and a spreadsheet with zonal lacunarity values as output. As a demonstration, lacunarity measurements of grasslands in Corinth and Lake Dallas, Texas were calculated from the 2001 and 2011 NLCD data using box sizes (scales) of 2, 3, 4, 5, 6, 7, 8, 9, and 10. The results show that measures of grassland lacunarity in Lake Dallas were higher than Corinth at all scales, and the measures of grassland lacunarity in 2011 were higher than 2001 for both cities because of the increasing gap sizes in grasslands. The increasing gap sizes in grasslands were caused by converting the grasslands into developed areas. The results suggest that the zonal lacunarity analysis tool can provide important information on the spatial distribution of gaps in the input discrete raster data at different scales. It is hoped that the zonal lacunarity analysis tool can be further evaluated using different datasets in landscape ecology.

Scale-dependent stratification: a skyline–shoreline scatter plot

Most performance exercises help identify size-dependent and size-independent indicators which separately represent quantity and quality proxies. A good example is the evaluation of a country’s scientific performance relative to its economic standing. Scientific performance is usually evaluated in terms of publications and citations and economic performance is measured in terms of gross domestic product (GDP). In this paper we use exergy, which is a scalar second-order product of publications and citations as a measure of output, with GDP as input. X and GDP are scale-dependent terms. We shall take the ratio X/GDP as the scale-dependent measure of performance relative to economic strength. A scatter plot of X/GDP to GDP serves as a quality–quantity map and from this it is possible to identify what are called the skyline and shoreline boundaries showing the scale-dependent upper and lower bounds of performance. One lesson that emerges is that quality does not necessarily grow with size; there is a noticeable scale-dependent stratification. It is also possible to identify a few economies that stand out noticeably from the rest.

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A numerical study of turbulence under temporally evolving axisymmetric contraction and subsequent relaxation

Direct numerical simulations using up to $4096^{3}$ grid points on a deforming domain have been used to study the response of initially isotropic turbulence to a period of spatially uniform axisymmetric contraction (with one extensional and two equally compressive directions) and subsequent relaxation. A time-dependent strain rate is formulated to closely correspond to the downstream evolution in the wind tunnel experiments of Ayyalasomayajula & Warhaft (J. Fluid Mech., vol. 566, 2006, pp. 273–307), with a smoothly varying 4 : 1 contraction ratio. The application of strain leads to anisotropy in both the large scales and the small scales, in a manner where nonlinear effects not considered in rapid-distortion theory play an important role. Upon termination of strain, the small scales quickly return to isotropy while a residual level of anisotropy appears to persist at the large scales. The simulations are shown to reproduce many key findings from experiments, including distinctive changes in the form of the one-dimensional spectra in the extensional direction that arise at sufficiently high Reynolds number, during both the straining and relaxation periods. Scale-dependent measures of anisotropy are presented in terms of one-dimensional spectra and axisymmetric versions of the energy spectrum. To explain the observed changes in spectral shapes, various terms in the spectral evolution equation representing rapid pressure strain, slow pressure strain, production, nonlinear transfer and viscous dissipation are computed, showing that nonlinear effects take a dominant role when a wide range of scales exists. In particular, the ‘double-peak’ spectral form observed in experiments at high Reynolds number is found to be a consequence of the small scales relaxing towards isotropy much faster than the large scales. A comparison of results obtained from computational domains of varying sizes and grid resolutions show that the numerical findings are robust.

Scale-dependent measurements of meteorite strength: Implications for asteroid fragmentation

Measuring the strengths of asteroidal materials is important for developing mitigation strategies for potential Earth impactors and for understanding properties of in situ materials on asteroids during human and robotic exploration. Studies of asteroid disruption and fragmentation have typically used the strengths determined from terrestrial analog materials, although questions have been raised regarding the suitability of these materials. The few published measurements of meteorite strength are typically significantly greater than those estimated from the stratospheric breakup of meter-sized meteoroids. Given the paucity of relevant strength data, the scale-varying strength properties of meteoritic and asteroidal materials are poorly constrained. Based on our uniaxial failure studies of centimeter-sized cubes of a carbonaceous and ordinary chondrite, we develop the first Weibull failure distribution analysis of meteorites. This Weibull distribution projected to meter scales, overlaps the strengths determined from asteroidal airbursts and can be used to predict properties of to the 100m scale. In addition, our analysis shows that meter-scale boulders on asteroids are significantly weaker than small pieces of meteorites, while large meteorites surviving on Earth are selected by attrition. Further, the common use of terrestrial analog materials to predict scale-dependent strength properties significantly overestimates the strength of meter-sized asteroidal materials and therefore is unlikely well suited for the modeling of asteroid disruption and fragmentation. Given the strength scale-dependence determined for carbonaceous and ordinary chondrite meteorites, our results suggest that boulders of similar composition on asteroids will have compressive strengths significantly less than typical terrestrial rocks.

Spatial Distribution Patterns in the Very Rare and Species-Rich Picea chihuahuana Tree Community (Mexico).

The very rare Mexican Picea chihuahuana tree community covers an area of no more than 300 ha in the Sierra Madre Occidental. This special tree community has been the subject of several studies aimed at learning more about the genetic structure and ecology of the species and the potential effects of climate change. The spatial distribution of trees is a result of many ecological processes and can affect the degree of competition between neighbouring trees, tree density, variability in size and distribution, regeneration, survival, growth, mortality, crown formation and the biological diversity within forest communities. Numerous scale-dependent measures have been established in order to describe spatial forest structure. The overall aim of most of these studies has been to obtain data to help design preservation and conservation strategies. In this study, we examined the spatial distribution pattern of trees in the P. chihuahuana tree community in 12 localities, in relation to i) tree stand density, ii) diameter distribution (vertical structure), iii) tree species diversity, iv) geographical latitude and v) tree dominance at a fine scale (in 0.25 ha plots), with the aim of obtaining a better understanding of the complex ecosystem processes and biological diversity. Because of the strongly mixed nature of this tree community, which often produces low population densities of each tree species and random tree fall gaps caused by tree death, we expect aggregated patterns in individual Picea chihuahuana trees and in the P. chihuahuana tree community, repulsive Picea patterns to other tree species and repulsive patterns of young to adult trees. Each location was represented by one plot of 50 x 50 m (0.25 ha) established in the centre of the tree community. The findings demonstrate that the hypothesis of aggregated tree pattern is not applicable to the mean pattern measured by Clark-Evans index, Uniform Angle index and Mean Directional index of the uneven-aged P. chihuahuana trees and P. chihuahuana tree community and but to specific spatial scales measured by the univariate L-function. The spatial distribution pattern of P. chihuahuana trees was found to be independent of patches of other tree species measured by the bivariate L-function. The spatial distribution was not significantly related to tree density, diameter distribution or tree species diversity. The index of Clark and Evans decreased significantly from the southern to northern plots containing all tree species. Self-thinning due to intra and inter-specific competition-induced mortality is probably the main cause of the decrease in aggregation intensity during the course of population development in this tree community. We recommend the use of larger sampling plots (> 0.25 ha) in uneven-aged and species-rich forest ecosystems to detect less obvious, but important, relationships between spatial tree pattern and functioning and diversity in these forests.

Scale-dependent homogeneity measures for causal dynamical triangulations

I propose two scale-dependent measures of the homogeneity of the quantum geometry determined by an ensemble of causal triangulations. The first measure is volumetric, probing the growth of volume with graph geodesic distance. The second measure is spectral, probing the return probability of a random walk with diffusion time. Both of these measures, particularly the first, are closely related to those used to assess the homogeneity of our own universe on the basis of galaxy redshift surveys. I employ these measures to quantify the quantum spacetime homogeneity as well as the temporal evolution of quantum spatial homogeneity of ensembles of causal triangulations in the well-known physical phase. According to these measures, the quantum spacetime geometry exhibits some degree of inhomogeneity on sufficiently small scales and a high degree of homogeneity on sufficiently large scales. This inhomogeneity appears unrelated to the phenomenon of dynamical dimensional reduction. I also uncover evidence for power-law scaling of both the typical scale on which inhomogeneity occurs and the magnitude of inhomogeneity on this scale with the ensemble average spatial volume of the quantum spatial geometries.

How Does Drifter Position Uncertainty Affect Ocean Dispersion Estimates?

AbstractTo develop methodologies to maximize the information content of Lagrangian data subject to position errors, synthetic trajectories produced by both a large-eddy simulation (LES) of an idealized submesoscale flow field and a high-resolution Hybrid Coordinate Ocean Model simulation of the North Atlantic circulation are analyzed. Scale-dependent Lagrangian measures of two-particle dispersion, mainly the finite-scale Lyapunov exponent [FSLE; λ(δ)], are used as metrics to determine the effects of position uncertainty on the observed dispersion regimes. It is found that the cumulative effect of position uncertainty on λ(δ) may extend to scales 20–60 times larger than the position uncertainty. The range of separation scales affected by a given level of position uncertainty depends upon the slope of the true FSLE distribution at the scale of the uncertainty. Low-pass filtering or temporal subsampling of the trajectories reduces the effective noise amplitudes at the smallest spatial scales at the expense of limiting the maximum computable value of λ. An adaptive time-filtering approach is proposed as a means of extracting the true FSLE signal from data with uncertain position measurements. Application of this filtering process to the drifters with the Argos positioning system released during the LatMix: Studies of Submesoscale Stirring and Mixing (2011) indicates that the measurement noise dominates the dispersion regime in λ for separation scales δ < 3 km. An expression is provided to estimate position errors that can be afforded depending on the expected maximum λ in the submesoscale regime.

Lunar surface roughness based on multiscale morphological method

Surface roughness is a useful tool to reflect numerous geological characteristics. Lunar Orbiter Laser Altimeter (LOLA) Gridded Data Records (GDRs) are used as the datum. In this paper, Lunar surface roughness maps are built based on morphological methods in image processing. As roughness measure, elevations of GDRs are considered as pixels of an image. Structuring element (SE) is employed as a scale-dependent measure of roughness maps. Global roughness maps with different resolutions are built to interpret the stability of our roughness measure. Global roughness with different-size SEs is mapped based on GDRs with the resolution of 64 pixels per degree to discuss the roughness variations in local regions determined by SEs. Regional roughness maps provide significant melt-related overviews of typical topography.

The 6dF Galaxy Survey: cosmological constraints from the velocity power spectrum

We present scale-dependent measurements of the normalised growth rate of structure $f\sigma_{8}(k, z=0)$ using only the peculiar motions of galaxies. We use data from the 6-degree Field Galaxy Survey velocity sample (6dFGSv) together with a newly-compiled sample of low-redshift $(z < 0.07)$ type Ia supernovae. We constrain the growth rate in a series of $\Delta k \sim 0.03 h{\rm Mpc^{-1}}$ bins to $\sim35\%$ precision, including a measurement on scales $>300 h^{-1}{\rm Mpc}$, which represents one of the largest-scale growth rate measurement to date. We find no evidence for a scale dependence in the growth rate, or any statistically significant variation from the growth rate as predicted by the {\it Planck} cosmology. Bringing all the scales together, we determine the normalised growth rate at $z=0$ to $\sim15\%$ in a manner {\it independent} of galaxy bias and in excellent agreement with the constraint from the measurements of redshift-space distortions from 6dFGS. We pay particular attention to systematic errors. We point out that the intrinsic scatter present in Fundamental-Plane and Tully-Fisher relations is only Gaussian in logarithmic distance units; wrongly assuming it is Gaussian in linear (velocity) units can bias cosmological constraints. We also analytically marginalise over zero-point errors in distance indicators, validate the accuracy of all our constraints using numerical simulations, and demonstrate how to combine different (correlated) velocity surveys using a matrix `hyper-parameter' analysis. Current and forthcoming peculiar velocity surveys will allow us to understand in detail the growth of structure in the low-redshift universe, providing strong constraints on the nature of dark energy.

Small-scale anisotropic intermittency in magnetohydrodynamic turbulence at low magnetic Reynolds numbers.

Small-scale anisotropic intermittency is examined in three-dimensional incompressible magnetohydrodynamic turbulence subjected to a uniformly imposed magnetic field. Orthonormal wavelet analyses are applied to direct numerical simulation data at moderate Reynolds number and for different interaction parameters. The magnetic Reynolds number is sufficiently low such that the quasistatic approximation can be applied. Scale-dependent statistical measures are introduced to quantify anisotropy in terms of the flow components, either parallel or perpendicular to the imposed magnetic field, and in terms of the different directions. Moreover, the flow intermittency is shown to increase with increasing values of the interaction parameter, which is reflected in strongly growing flatness values when the scale decreases. The scale-dependent anisotropy of energy is found to be independent of scale for all considered values of the interaction parameter. The strength of the imposed magnetic field does amplify the anisotropy of the flow.